T-6

Goal: Trap model. Captures aging in a simplified single particle description.

Key concepts: gradient descent, rugged landscapes, metastable states, Hessian matrices, random matrix theory, landscape’s complexity.

Langevin, Activation

- Monte Carlo dynamics. Consider the REM discussed in Problems 1. Assume that the $2^{N}$ configurations are organised in an hypercube of connectivity $N$: each configuration has $N$ neighbours, that are obtained flipping one spin of the first configuration. -Langevin dynamics -Arrhenius law, trapping and activation.

The energy landscape of the REM.

-aging

Problem 6.1: from the REM to the trap model

1. The golf course energy landscape. The smallest energies values $E_{\alpha }$ among the $M=2^{N}$, those with energy density $\epsilon =-{\sqrt {\log 2}}$, can be assumed to be distributed as Failed to parse (syntax error): {\displaystyle P_N(E) \approx C_N \text{exp}\left[2 \sqrt{\log 2}(E+ N \sqrt{ \log 2} )} \right] } , where $C_{N}$ is a normalisation factor. Justify the form of this distribution (Hint: recall the discussion on extreme value statistics in Lecture 1!). Consider now one of these deep configurations of very small energy, which we will call <em< traps : what is the minimal energy density among the neighbouring configurations of the trap? Does it depend on the entry of the trap itself? Why is this consistent with the results on the entropy of the REM that we computed in Problem 1.1?


2. Trapping times. The results above show that the energy landscape of the REM has a "golf course" structure: configuration with deep energy are isolated, surrounded by configurations of much higher energy (zero energy density). The Arrhenius law states that the time needed for the system to escape from a trap of energy $E$ and reach a configuration of zero energy is $\tau \sim e^{-\beta E}$. We call this a trapping time . Given the energy distribution $P_{N}(E)$, determine the distribution of trapping times $P(\tau )$. Show that there is a critical temperature below which the average trapping time diverges, and therefore the system needs infinite time to explore the whole configuration space. Do you recognise this temperature?

The trap model is an effective model for the dynamics in complex landscapes. In the model, the configuration space is described as a collection of $M=2^{N}$ traps with depths (energies) distributed as $P_{N}(E)$, and the dynamics is a random walk between the traps: the system is trapped in one configuration for a random time distributed as $P(\tau )$, then it exits the traps and falls into another one randomly chosen.

1. Aging . Consider a trap-like dynamics from $t=0$ to $t>0$. Compute the typical value of the maximal trapping time $\tau _{\text{max}}(t)$ encountered in this time interval, an show that below the critical temperature $\tau _{\text{max}}(t)\sim t$. Who si this a condensation phenomenon, as the ones discussed in Problems 1? Why this corresponds to aging?


2. .

the system has spent almost all the time up to $t$ in only one trap, the deepest one.